Problem: $\dfrac{ 7x + 4y }{ 6 } = \dfrac{ 5x - 2z }{ 2 }$ Solve for $x$.
Multiply both sides by the left denominator. $\dfrac{ 7x + 4y }{ {6} } = \dfrac{ 5x - 2z }{ 2 }$ ${6} \cdot \dfrac{ 7x + 4y }{ {6} } = {6} \cdot \dfrac{ 5x - 2z }{ 2 }$ $7x + 4y = {6} \cdot \dfrac { 5x - 2z }{ 2 }$ Reduce the right side. $7x + 4y = {6} \cdot \dfrac{ 5x - 2z }{ {2} }$ $7x + 4y = {3} \cdot \left( 5x - 2z \right)$ Distribute the right side $7x + 4y = {3} \cdot \left( {5x} - {2z} \right)$ $7x + 4y = {15}x - {6}z$ Combine $x$ terms on the left. ${7x} + 4y = {15x} - 6z$ $-{8x} + 4y = -6z$ Move the $y$ term to the right. $-8x + {4y} = -6z$ $-8x = -6z - {4y}$ Isolate $x$ by dividing both sides by its coefficient. $-{8}x = -6z - 4y$ $x = \dfrac{ -6z - 4y }{ -{8} }$ All of these terms are divisible by $2$ Divide by the common factor and swap signs so the denominator isn't negative. $x = \dfrac{ {3}z + {2}y }{ {4} }$